Help with a MOSFET exercise

I must admit I'm a noob when it comes to Analog Circuits, I need help!!! it actually took me a while to understand what the problem is and how to explain it to you guys.

I'm given the attached exercise and I'm asked to calculate voltage gain and output resistance, both with written computations and with a simulation of the circuit in MicroCap. The purpose is basically to compare the values and discuss possible discrepancies.

I have two questions about the calculation of output resistance. It seems like an easy task but I think the consultation I had not long ago with my teacher confused me even more.

I found this equation:

Code ( (Unknown Language)):

Rout = 1 / (λ * Kp (Ugs - Ut)^2) = 1 / (lambda * Id)

I also found that when W and L are given

Code ( (Unknown Language)):

Id = 1/2 * Kp * W / L * (Ugs - Uth)^2

My first confusion and question is that if λ = 0, as you can see in the picture, it automatically means that Rout goes off to infinity, but why would my teacher give me such an exercise to compute?

Second question. Assuming that λ is different from 0. I've been trying to find 'Rout' with the two equations I know but there's always something wrong while solving. I don't know how to find Id. My teacher told me that

Code ( (Unknown Language)):

Ugs = (5 - Rs * Id)

but then if

Code ( (Unknown Language)):

Id = 1/2 * Kp * W / L * (Ugs - Uth)^2

and

Code ( (Unknown Language)):

Ugs = (5 - Rs * Id)

Does the equation above make sense? why do I have two Id's on both sides of the equation? assuming this is right, the result for 'Id' I get does not make sense.

Another confusion is that in the exercise C3 is not given. In the MicroCap simulation this capacitor, as you may know, regulates the value of 'Id' but then I get totally confused because Id depends on C3 but C3 is nowhere in the equations. I hope this is clear and I'm sorry if the answer is too simple.

If there's anybody who can help me clarify this I'd really appreciate it. I really struggle with this course badly, this is for sure because of my poor background.